Common Logarithm — Characteristic and Mantissa

For numerical work the common logarithm (base 10) is used, and its value is split into two parts:

• The characteristic is the integral (whole-number) part of the logarithm. It may be positive, zero or negative, and it fixes only the position of the decimal point in the number.
• The mantissa is the decimal (fractional) part. It is always kept positive (lying between 0 and 1) and depends only on the sequence of significant digits of the number, never on where the decimal point sits.

The key insight — the one that makes a single log table usable for numbers of every size — is that numbers with the same significant digits have the same mantissa and differ only in their characteristic. Thus $\log 4567$, $\log 45.67$ and $\log 0.4567$ all share the mantissa of the digit string "4567"; only their characteristics differ.

Rule for the characteristic.

Case 1 — number greater than or equal to 1. The characteristic is one less than the number of digits in the integral part:

So $\log 4567$ has characteristic $4-1 = 3$; $\log 45.67$ has characteristic $2-1 = 1$; $\log 4.567$ has characteristic $1-1 = 0$.

Case 2 — number less than 1. The characteristic is negative, equal to (number of zeros immediately after the decimal point, before the first non-zero digit) plus one, and is written with a bar over the digit to show that only the characteristic is negative while the mantissa stays positive:

So $\log 0.4567$ has characteristic $\bar{1}$ (i.e. $-1$, no leading zeros); $\log 0.04567$ has characteristic $\bar{2}$ ($-2$, one leading zero); $\log 0.004567$ has characteristic $\bar{3}$ ($-3$, two leading zeros). …